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Full text of "Treatise On Analysis Vol-Ii"

276       XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

(ii) If the sequences (A, \i) and (|A| * |/i|, v) are convolvable > then so is
(A, ju, v). Likewise if the sequences (X v) and (A, |/x| * |v|) are convolvable.

We may restrict ourselves to the case in which A, ^, and v are positive.
Suppose that the sequence (A, /*, v) is convolvable; then for every compact
subset K of G the set of triples (x, y, z) such that xyz e K is (A  ^  v)-
integrable. Let A be the set of pairs (x, y) such that xy e K. For each compact
subset K' of G, the set A x K' c: G3 is contained in the set of triples (x, y, z)
such that xyz e KK', and since KK' is compact, it follows that A x K' is
((A  ju)  v)-integrable. Since v ^ 0, this implies that A is (A  ^)-integrable
(13.21.11) and hence that (A, //) is convolvable. Consequently, for any
f e JT+(G), it follows from the hypothesis and the Lebesgue-Fubini theorem
that

f *dv(z) (*f(tz) d(X * A0(0 = f *dv(z)

since the function t\-*f(tz) is in Jf+(G), for each x e G. This shows that A * ^
and v are convolvable. One proves in the same way that (/*, v) and (A, n * v)
are convolvable. The formula (14.7.2.1) is then a consequence of the Lebesgue-
Fubini theorem.

Conversely, suppose that (A, //) and (A * /*, v) are convolvable, and let /
be a function belonging to #"+(G). For each zeG, the function t\-*f(tz)
belongs to Jf*4.(G), hence is (A * ^)-integrable, and we have

= ||/(xyz)^0

Hence, by Lebesgue-Fubini, it follows that
**

) f

which proves that (A, ju, v) is convolvable.

One can give examples of measures A, ju, v on G such that the pairs (A, /*),
(A * fi9 v), (fi, v) and (A, /* * v) are convolvable but (A * /*) * v ^ A * (/i * v)
(Problem 1).G.